where \(a_{n}\), \(b_{n}\), \(\alpha _{n}\), \(\beta _{n}\), \(n\in \mathbb {N}_{0}\), and initial values \(x_{-i}\), \(y_{-i}\), \(i\in\{1,2\}\) are real numbers, are found. The domain of undefinable solutions to the system is described. The long-term behavior of its solutions is studied in detail for the case of constant \(a_{n}\), \(b_{n}\), \(\alpha _{n}\) and \(\beta _{n}\), \(n\in \mathbb {N}_{0}\).

Studying concrete nonlinear difference equations and systems is a topic of a great recent interest (see, e.g., [1–46] and the references therein). Studying systems of difference equations, especially symmetric and close to symmetric ones, is a topic of considerable interest (see, e.g., [2, 6, 7, 10, 12–16, 18, 19, 23, 24, 26–29, 31–38, 40, 41, 44, 46]). Another topic of interest is solvable difference equations and systems and their applications (see, e.g., [1–5, 7, 17, 20, 21, 23–27, 29–37, 39–46]). Renewed interest in the area started after the publication of [20] where a formula for a solution of a difference equation was theoretically explained. The most interesting thing in [20] was a change of variables which reduced the equation to a linear one with constant coefficients. Related ideas were later used, e.g., in [1, 4, 7, 17, 21, 23–27, 29–37, 39–45].

Quite recently in [2] the following systems of difference equations were presented:

where \(a_{n}\), \(b_{n}\), \(\alpha _{n}\), \(\beta _{n}\), \(n\in \mathbb {N}_{0}\), and initial values \(x_{-i}\), \(y_{-i}\), \(i\in\{1,2\}\), are real numbers, is a generalization of the system in (1). Our aim is to show that more general system (2) is solvable by giving a natural method for getting its solutions. The domain of undefinable solutions to the system is also described. For the case when \(a_{n}\), \(b_{n}\), \(\alpha _{n}\), \(\beta _{n}\), \(n\in \mathbb {N}_{0}\), are constant, the long-term behavior of its solutions is investigated in detail.

A solution \((x_{n}, y_{n})_{n\ge-2}\) of system (2) is called periodic, or eventually periodic, with period p if there is \(n_{0}\ge-2\) such that

Assume first that \(x_{-i}\ne0\), \(y_{-i}\ne0\), \(i\in\{1,2\}\). Then, by the method of induction and the equations in (2), it follows that for every well-defined solution to system (2), \(x_{n}\ne0\) and \(y_{n}\ne0\), for every \(n\in \mathbb {N}_{0}\). On the other hand, if \(x_{n_{0}}=0\) for some \(n_{0}\in \mathbb {N}\), then the first equation in (2) implies that \(y_{n_{0}-1}=0\) or \(y_{n_{0}-2}=0\). If \(y_{n_{0}-1}=0\), then \(x_{n_{0}-2}=0\) or \(x_{n_{0}-3}=0\), while if \(y_{n_{0}-2}=0\), then \(x_{n_{0}-3}=0\) or \(x_{n_{0}-4}=0\). Repeating this procedure, we get that \(x_{-i}=0\) or \(y_{-i}=0\) for some \(i\in\{1,2\}\). Similarly, if \(y_{n_{1}}=0\) for some \(n_{1}\in \mathbb {N}\), we get \(x_{-i}=0\) or \(y_{-i}=0\) for some \(i\in\{1,2\}\). Hence, for a well-defined solution \((x_{n},y_{n})_{n\ge-2}\) of system (2), we have that

$$\begin{aligned} x_{n}y_{n}\ne0,\quad n\ge-2 \end{aligned}$$

(3)

if and only if \(x_{-i}y_{-i}\ne0\), \(i\in\{1,2\}\).

Assume now that \((x_{n},y_{n})_{n\ge-2}\) is a solution to system (2) such that (3) holds. Then, by multiplying the first equation in (2) by \(x_{n-1}\) and the second one by \(y_{n-1}\), and using the following changes of variables

Before we formulate and prove the main results regarding the long-term behavior of well-defined solutions to system (14), we quote the following well-known asymptotic formula which will be used in the proofs of the main results:

Letting \(m\to\infty\) in (38) and (39) and using the condition \(\vert a\alpha \vert >1\), we have \(p_{m}\to0\) and \(\vert \hat{p}_{m}\vert \to\infty\), from which along with (22) and (23) the statement easily follows.

(e) By using the condition \(u_{-1}\ne(a\beta +b)/(1-a\alpha )=u_{0}\), we get

Letting \(m\to\infty\) in (40) and (41) and using the condition \(\vert a\alpha \vert >1\), we have \(\vert p_{m}\vert \to\infty\) and \(\hat{p}_{m}\to0\), from which along with (22) and (23) the statement easily follows.

(f) By using the condition \(v_{-1}=(a\beta +b)/(1-a\alpha )\ne v_{0}\), we get

Letting \(m\to\infty\) in (42) and (43) and using the condition \(\vert a\alpha \vert >1\), we have \(q_{m}\to0\) and \(\vert \hat{q}_{m}\vert \to\infty\), from which along with (24) and (25) the statement easily follows.

(g) By using the condition \(v_{-1}\ne(a\beta +b)/(1-a\alpha )= v_{0}\), we get

Letting \(m\to\infty\) in (44) and (45) and using the condition \(\vert a\alpha \vert >1\), we have \(\vert q_{m}\vert \to\infty\) and \(\hat{q}_{m}\to0\), from which along with (24) and (25) the statement easily follows.

Acknowledgements

The work of the first and the second authors was supported by the Serbian Ministry of Education and Science, project III 41025. The work of the first author was also supported by the Serbian Ministry of Education and Science, project III 44006. The work of the second author was also supported by the Serbian Ministry of Education and Science, project OI 171007. The work of the third author was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by project CZ.1.05/1.1.00/02.0068 financed from the European Regional Development Fund. The third author was also supported by the project FEKT-S-14-2200 of Brno University of Technology.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.